O problema não vai sumir sem que alguém faça alguma coisa.

Vai sumir is the very common spoken way to express the future in Brazilian Portuguese:

• vai + infinitive = is going to / will
• vai sumir = will disappear / is going to go away

So:

• O problema não vai sumir... = The problem won’t go away...

You could also say:

• O problema não sumirá...

That is the simple future, but in everyday Brazilian Portuguese, ir + infinitive is usually more natural in conversation.

Here, sumir means to disappear, to go away, or to stop being there.

With problema, English often translates it more naturally as:

• The problem won’t go away

rather than the more literal:

• The problem won’t disappear

Portuguese also has desaparecer, which can mean to disappear, but sumir is very common in everyday speech and sounds natural here.

In standard Portuguese, não normally goes before the conjugated verb.

So:

• não vai sumir = correct
• vai não sumir = not standard in this sentence

Because vai is the conjugated verb, the negation comes before it:

• não vai
• não pode
• não quer
• não fez

This is one of the most basic word-order patterns in Portuguese.

Sem que means without ...ing or without the fact that ... happens, when what follows is a full clause with its own subject.

In this sentence:

• sem que alguém faça alguma coisa
• literally: without someone doing something

Portuguese often uses:

• sem + infinitive when there is no new subject
• sem que + subjunctive when a new subject is introduced

Compare:

• Ele saiu sem falar.
  He left without speaking.
  Same subject: he

• O problema não vai sumir sem que alguém faça alguma coisa.
  The problem won’t go away unless someone does something / without someone doing something.
  New subject: alguém

Because sem que normally requires the subjunctive.

So:

• faça = present subjunctive of fazer
• faz = present indicative

After sem que, Portuguese expects the subjunctive because the action is not presented as a simple fact. It is hypothetical, uncertain, or dependent.

Examples:

• sem que ele saiba = without him knowing
• sem que ela veja = without her seeing
• sem que alguém faça alguma coisa = without someone doing something

So sem que alguém faz would sound wrong in standard Portuguese.

Usually, for standard and natural Portuguese, no.

When the clause has its own subject, Portuguese strongly prefers:

• sem que + subjunctive

So this is the natural structure:

• sem que alguém faça alguma coisa

Using sem + infinitive works best when the subject stays the same:

• Ele saiu sem dizer nada.
• Ela entrou sem bater.

Trying to insert a new subject directly after sem is much less natural here.

Alguém means someone or anyone, depending on context.

Here it means:

• someone has to do something
• the problem will not go away unless someone acts

So the idea is not about a specific person, but about at least one person taking action.

It is an indefinite subject, which is very common in Portuguese.

Alguma coisa means something.

So:

• fazer alguma coisa = do something

This is a very common everyday expression in Brazilian Portuguese.

You could also say:

• fazer algo

But algo is a bit more formal or written-sounding than alguma coisa.

Why not nada?

• fazer nada means do nothing
• That would change the meaning completely

So:

• alguém faça alguma coisa = someone does something
• alguém faça nada = not correct in standard Portuguese
• ninguém faça nada = nobody does anything

That is a great question, because Portuguese does not always match English here.

English often prefers:

• The problem won’t go away unless someone does something
• The problem won’t go away without anyone doing anything

Portuguese can still use alguma coisa naturally in this kind of structure:

• sem que alguém faça alguma coisa

Even though the main clause is negative (não vai sumir), the subordinate clause is still naturally expressed with alguém and alguma coisa.

A more heavily negative version is also possible:

• sem que ninguém faça nada

That means roughly:

• without anybody doing anything

Both patterns exist, but the sentence you were given is very normal and straightforward.

Portuguese usually needs an article before a noun in cases like this.

So:

• O problema = the problem

Leaving out the article would sound unnatural in a normal sentence:

• Problema não vai sumir... = not natural in standard speech

Portuguese uses articles more often than English does. If a specific problem is being discussed, o problema is exactly what you would expect.

They are very close in meaning, but not identical in structure.

• sem que alguém faça alguma coisa
  literally: without someone doing something

• a menos que alguém faça alguma coisa
  unless someone does something

In this sentence, English would often translate the Portuguese more naturally with unless:

• The problem won’t go away unless someone does something.

So while the Portuguese literally uses sem que, the overall meaning is very close to unless.

Faça is the present subjunctive of fazer.

Here is the present subjunctive of fazer:

• eu faça
• você/ele/ela faça
• nós façamos
• vocês/eles/elas façam

In your sentence:

• alguém faça
• someone do / someone does in a subjunctive environment

This is one of the most important subjunctive patterns to learn, because triggers like sem que, para que, antes que, and embora are very common.

It is basically neutral, natural Brazilian Portuguese.

• vai sumir sounds conversational and everyday
• sem que alguém faça alguma coisa is standard grammar
• the whole sentence sounds completely normal in speech and writing

If you wanted a slightly more formal version, you might see something like:

• O problema não desaparecerá sem que alguém tome alguma providência.

But your original sentence is very natural and useful for real-life Portuguese.